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Projective geometry

In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of homogeneous co-ordinates it looks like an extension or technical improvement of the use of co-ordinates to reduce geometric problems to algebra, that reduced the number of special cases. And on the other hand the detailed study of quadrics and the 'line geometry' of Julius Plucker still forms a rich set of examples for geometers who also work with more general concepts. Towards the end of the century the Italian school (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.

It should be said first that the notable projective geometers, including Poncelet, Steiner and others, were not intending to extend analytic geometry. Techniques were supposed to be synthetic: in effect projective space as we understand it now was to be introduced on an axiomatic basis. This poses some problems in recovering the theory. In the case of the projective plane alone, the axiomatic approach may encounter models that cannot be described via linear algebra.

Whatever the precise foundational status, projective geometry did include basic incidence properties. That means that any two distinct lines L and M in the projective plane intersect in exactly one point P. The special case in analytic geometry of parallel lines has been subsumed in the smoother form of a line at infinity on which P will lie in that case. The point is then that the line at infinity is a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).

It also included a full theory of conic sections, a subject that already had a large number of theorems (mainly useful as a source of examination questions). There are great and clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity; and that a parabola is tangent to the same line. The whole family of circles can be seen as the conics passing through two given points on the line at infinity – at the cost of allowing complex number co-ordinates. Since co-ordinates were not 'synthetic', one replaces that by fixing a line and two points on it, and considering the linear system of all conics passing through those points as the basic object of study. This whole approach was very attractive to talented geometers, and the field was thoroughly worked over. A later many-volume work by H.F. Baker shows the style.

This period in geometry was rather overtaken by the research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. In the latter part of the nineteenth century, the detailed study of projective geometry itself was less important to professional mathematicians, though the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now considered an anticipation of the theory of Chern classes in their guise as representing the algebraic topology of Grassmannians.

See also

References

  • Coxeter, H. S. M., The Real Projective Plane, 3rd ed, Springer Verlag, New York, 1995
  • Coxeter, H. S. M., Projective Geometry, 2nd ed., Springer Verlag, New York, 2003
  • Veblen, O. and Young, J. W., Projective Geometry, 2 vols., Blaisdell Pub. Co., New York, 1938–46

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